Ninja edit after looking at the links a little more:
I should have specified dark grapes, or maybe muscat grapes? I remember liking both kinds in my last trip to Europe. The taste of the Welch white grape is pretty bland. I think they use it to sweeten other juices because of that.
It’s sort of circularly defined as it’s the taste of Concord grapes (and Catawba.) I happen to like it, more in actual grapes and wine (!)* than in grape juice per se, in which it can indeed be a bit astringent-tasting and overpowering.
*Although I don’t like the typical Concord wine’s sweetness.
“Foxiness” is the characteristic taste of Labrusca native American grapes, such as Concord grapes (from which most grape juice in the US is made – even “White” grape juice. ) , but also including other varieties like Catawba, Niagara, Diamond, and others. The name of the taste comes from the “Fox” grape, which is, I believe, from which the others derive (Concord, though common, is a bred variety, developed in Concord, Massachusetts. Duh.)*
Taste is difficult to describe except by analogy, but most grape jellies, jams, and juices (including Welch’s and Smuckers) are made from such grapes, and you can sort of define the “foxy” flavor as the difference between the way those grapes and grape products taste and the way , say California “table grapes” and other seedless snack grapes taste.
It also shows up as the difference between wines made from Labrusca grapes and European vinifera grapes. It shows up in, for instance, Manischevitz and Mogen David wine. Most people don’t care for foxiness in their wine. I’m one of the ones that do like it, and I like Widmer’s Lake Niagara wine and several of the Bully Hill wines. Labrusca wines need not be sweet or even fruity (a lot of the Bully Hill wines aren’t), but that’s how people tend to characterize them. (Including Pepper Mill, who loathes my Widmer’s wine. More for me!)
Ever since I’ve read Lewis Carroll discoursing on the etymology of “fox” in many place and plant names, I’ve come to suspect it might be at the root of the name of the fox grape. as Carroll remarked, the “fox” in “foxglove” doesn’t refer to a furry canine-;like beast, but is a corruption of “Folk” – as in “Fairy Folk”. Certainly that explains “Foxfire”, wood that has become luminous from the phosphorescent fungi growing on it. So the Fax Grape might be called that because it was associated or attributed to Fairies. Or maybe foxes ate it. I don’t know.